The Geometry of Time-Stepping

The space-time geometric structure of Maxwell’s equations is examined and a subset of them is found to define a pair of exact discrete time-stepping relations. The desirability of adopting an approach to the discretization of electromagnetic problems which exploits this fact is advocated, and the name topological time-stepping for numerical schemes complying with it is suggested. The analysis of the equations leading to this kind of time-stepping reveals that these equations are naturally written in terms of integrated field quantities associated with space-time domains. It is therefore suggested that these quantities be adopted as state variables within numerical methods. A list of supplementary prescriptions for a discretization of electromagnetic problems suiting this philosophy is given, with particular emphasis on the necessity to adopt a space-time approach in each discretization step. It is shown that some existing methods already comply with these tenets, but that this fa...

• Type: Journal
• Year: 2001
• Where: Progress in Electromagnetics Research
• Authors: Claudio Mattiussi
• Keywords: Finite Difference Time Domain, Finite Differences, Finite Element Methods, Finite Volume Methods